Page 469 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

456 Random Vibrations Chap. 13

13-26 Determine the complex form of the Fourier series for the rectangular wave shown in
Fig. P13-13 and plot its spectral density.
13-27 The sharpness of the frequency-response curve near resonance is often expressed in
terms oí Q = Points on either side of resonance where the response falls to a
value 1 / \/2 are called half-power points. Determine the respective frequencies of
the half-power points in terms of and Q.
13-28 Show that
drj
= ^ for f « 1
0 ( 1 - ri^y + (2^7))^
13-29 The differential equation of a system with structural damping is given as
mx -I- k(l -h iy)x = F(t)
Determine the frequency-response function.
13-30 A single-DOF system with natural frequency co„ and damping factor ¿' = 0.10 is
excited by the force
F(t) = Fcos(0.5io„/ - 0j) + Fcos((o^t - 62) + Fcos(2w^t - O^)
Show that the mean square response is
(1.74 + 25.0 + 0.11 0)i^ |^ |

13-31 In Example 13.7-3, what is the probability of the instantaneous acceleration exceed­
ing a value 53.2g? Of the peak value exceeding this value?
13-32 A large hydraulic press stamping out metal parts is operating under a series of forces
approximated by Fig. P13-32. The mass of the press on its foundation is 40 kg and its
natural frequency is 2.20 Hz. Determine the Fourier spectrum of the excitation and
the mean square value of the response.
T
4
— I I-

r
10® N h rh rh

= J s Figure P13-32.
13-33 For a single-DOF system, the substitution of Eq. (13.8-10) into Eq. (13.8-6) results in
df

where is the spectral density of the excitation force. When the damping is